5.5. Inequalities in Triangles http://www.ck12.org
Solution:Let’s start with 4 DCE. The missing angle is 55◦. By Theorem 5-9, the sides, in order areCE,CD, and
DE.
For 4 BCD, the missing angle is 43◦. Again, by Theorem 5-9, the order of the sides isBD,CD, andBC.
By the SAS Inequality Theorem, we know thatBC>DE, so the order of all the sides would be:BD=CE,CD,DE,BC.
SSS Inequality Theorem(also called the Converse of the Hinge Theorem)
SSS Inequality Theorem:If two sides of a triangle are congruent to two sides of another triangle, but the third side
of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is
greater in measure than the included angle of the second triangle.
Example 6:IfX Mis a median of 4 XY ZandXY>X Z, what can we say aboutm^6 1 andm^6 2? What we can deduce
from the following diagrams.
Solution:By the definition of a median,Mis the midpoint ofY Z. This means thatY M=MZ.MX=MXby the
Reflexive Property and we know thatXY>X Z. Therefore, we can use the SSS Inequality Theorem to conclude that
m^61 >m^6 2.
Example 7:List the sides of the two triangles in order, from least to greatest.
Solution:Here we have no congruent sides or angles. So, let’s look at each triangle separately. Start with 4 XY Z.
First the missing angle is 42◦. By Theorem 5-9, the order of the sides isY Z,XY, andX Z. For 4 W X Z, the missing
angle is 55◦. The order of these sides isX Z,W Z, andW X. Because the longest side in 4 XY Zis the shortest side in
4 W X Z, we can put all the sides together in one list:Y Z,XY,X Z,W Z,W X.
Example 8:Below is isosceles triangle 4 ABC. List everything you can about the triangle and why.